Geodesic
The Product of Two Legendre Polynomials
Glasgow mathematical journal, Tome 1 (1953) no. 3, pp. 121-125

Voir la notice de l'article provenant de la source Cambridge University Press

1. It is known that any polynomial in μ. can be expanded as a linear function of Legendre polynomials [1]. In particular, we haveThe earlier coefficients, say A0, A2, A4 may easily be found by equating the coefficients of μp+q, μp+q-2, μp+q-4 on the two sides of (1). The general coefficient A2k might then be surmised, and the value verified by induction. This may have been the method followed by Ferrers, who stated the result as an exercise in his Spherical Harmonics (1877). A proof was published by J. C. Adams [2]. The proof now to be given follows different lines from his. 
Dougall, John. The Product of Two Legendre Polynomials. Glasgow mathematical journal, Tome 1 (1953) no. 3, pp. 121-125. doi: 10.1017/S2040618500035590
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[1] 1.MacRobert, T. M., Spherical Harmonics (London), p. 95. Google Scholar

[2] 2.Adams, J. C., Proc. Roy. Soc., XXVII, 1878, p. 63; also Collected Scientific Papers, I, p. 187. Google Scholar

[3] 3.Hobson, E. W., Spherical and Ellipsoidal Harmonics (Cambridge), p. 83. Google Scholar

[4] 4.Bailey, W. N., “On the Product of Two Legendre Polynomials,” Proc. Camb. Phil. Soc., XXIX (1933), pp. 173–177. Google Scholar | DOI

[5] 5.Hardy, G. H., “A Chapter from Ramanujan's Notebook,” Proc. Camb. Phil. Soc., XXI (1923), pp. 492–503. Google Scholar

[6] 6.Dougall, J., “On Vandermonde's Theorem, and some general Expansions,” Proc Edin. Math. Soc., XXV (1907), pp. 114–132. Google Scholar

[7] 7.Bailey, W. N., Generalized Hypergeometric Series (Cambridge University Tract, 1935), Chaps. IV, V, VI. Google Scholar

[8] 8.Hardy, G. H., loc. cit., p. 496. Google Scholar

[9] 9.Dougall, J., loc cit., equation (10). Google Scholar

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